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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写宇宙学cosmology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宇宙学cosmology代写方面经验极为丰富，各种代写宇宙学cosmology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|宇宙学代写cosmology代考|The homogeneous Boltzmann equation revisited

Suppose that we are interested in the number density $n_1$ of species 1 . For simplicity, we will assume that the only process affecting the abundance of this species is a reaction with species 2 producing two particles, imaginatively called 3 and 4 . Schematically, $1+2 \leftrightarrow$ $3+4$; i.e., particle 1 and particle 2 can annihilate producing particles 3 and 4 , or the inverse process can produce 1 and 2. The Boltzmann equation for this system in an expanding universe was derived in Sect. 3.2.2, and the corresponding collision term in Sect. 3.2.3. Combining the general results Eq. (3.43) and Eq. (3.48), we obtain the following evolution equation for $n_1$ :

\begin{aligned} a^{-3} \frac{d\left(n_1 a^3\right)}{d t}=& \int \frac{d^3 p_1}{(2 \pi)^3 2 E_1} \int \frac{d^3 p_2}{(2 \pi)^3 2 E_2} \int \frac{d^3 p_3}{(2 \pi)^3 2 E_3} \int \frac{d^3 p_4}{(2 \pi)^3 2 E_4} \ & \times(2 \pi)^4 \delta_{\mathrm{D}}^{(3)}\left(p_1+p_2-p_3-p_4\right) \delta_{\mathrm{D}}^{(1)}\left(E_1+E_2-E_3-E_4\right)|\mathcal{M}|^2 \ & \times\left{f_3 f_4\left[1 \pm f_1\right]\left[1 \pm f_2\right]-f_1 f_2\left[1 \pm f_3\right]\left[1 \pm f_4\right]\right} \end{aligned}
Here, $E_i$ stands for $E_i\left(p_i\right)$ and $f_i$ for $f_i\left(p_i, t\right)$. We have thus obtained an integrodifferential equation for the phase-space distributions. Further, in principle at least, it must be supplemented with similar equations for the other species. In practice, these formidable obstacles can be overcome for many practical cosmological applications. The first, most important, realization is that scattering processes typically enforce kinetic equilibrium. That is, scattering takes place so rapidly that the distributions of thé various spécies take on the generic Bose-Einstein/Fermi-Dirac forms (Eq. (2.65) and Eq. (2.66)) with equal temperature $T$ for each species. This form condenses all of the freedom in the distribution into the functions of time $T$ and $\mu$. If annihilations were also in equilibrium, the sum of the chemical potentials $\mu_i$ in any reaction would have to balance. For example, the reaction $e^{+}+e^{-} \leftrightarrow \gamma+\gamma$ would lead to $\mu_{e^{+}}+\mu_{e^{-}}=2 \mu_\gamma$. In the out-of-equilibrium cases we will study, the system will not be in chemical equilibrium and we will have to solve a differential equation for $\mu$. The great simplifying feature of kinetic equilibrium, though, is that this differential equation will be a single ordinary differential equation, as opposed to the very complicated form of Eq. (4.1).

We will typically be interested in systems at temperatures smaller than $E-\mu$. In this limit, the exponential in the Bose-Einstein and Fermi-Dirac distributions is large and dwarfs the $\pm 1$ in the denominator. Thus, another simplification emerges: we can ignore the complications of quantum statistics, and the distributions follow the Boltzmann distribution of a classical dilute gas:
$$f(E) \rightarrow e^{\mu / T} e^{-E / T}$$

## 物理代写|宇宙学代写cosmology代考|Big Bang nucleosynthesis

Of the various epochs in the early universe, we have seen in Ch. 1 that Big Bang Nucleosynthesis (BBN) is of particular importance, as it produced the light elements we see in the universe and can be used to constrain cosmology. BBN happened when the temperature of the universe cooled to $1 \mathrm{MeV}$. At that point in time, the cosmic plasma consisted of:

• Relativistic particles in equilibrium: photons, electrons and positrons. These were kept in close contact with each other by electromagnetic interactions such as $e^{\prime} e \leftrightarrow$ $\gamma \gamma$. Besides a small difference due to fermion/boson statistics, these all had the same ahundances.
• Decoupled relativistic particles: neutrinos. $\Lambda \mathrm{t}$ temperatures a little above $1 \mathrm{MeV}$, the rate for processes such as $v e \leftrightarrow v e$ that keep neutrinos coupled to the rest of the plasma dropped beneath the expansion rate. Neutrinos therefore share the same temperature as the other relativistic particles (but see Sect. 2.4.4), and hence are roughly as abundant, but they do not couple to them.
• Nonrelativistic particles: baryons. If there had been no asymmetry in the initial number of baryons and anti-baryons, then both would be completely depleted by $1 \mathrm{MeV}$. However, such an asymmetry has to exist, since otherwise we would observe a universe almost completely devoid of baryons. Comparing the abundance of baryons to photons, we find $n_{\mathrm{b}} / s \sim 10^{-10}$ today. ${ }^1$ Since this ratio remains constant throughout the expansion (as long as the baryon number density is conserved), this also quantifies the baryon-antibaryon asymmetry in the early universe. As you can compute in Exercise $4.6$,
$$\eta_{\mathrm{b}} \equiv \frac{n_{\mathrm{b}}}{n_\gamma}=6.0 \times 10^{-10}\left(\frac{\Omega_{\mathrm{b}} h^2}{0.022}\right) .$$
There are thus many fewer baryons than relativistic particles in the universe.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|齐次玻尔兹曼方程重访

$$\eta_{\mathrm{b}} \equiv \frac{n_{\mathrm{b}}}{n_\gamma}=6.0 \times 10^{-10}\left(\frac{\Omega_{\mathrm{b}} h^2}{0.022}\right) .$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师